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\title{\vspace{-3.5cm}\textbf{The Report of Finite Element Method}}

\author{\textbf{何嘉兴}}
\date{}

\pagestyle{empty}

\begin{document}	
	
	\maketitle
	
\section{问题}
$$
-\Delta u = 2 \sin x \sin y,  u|_b = \sin x \sin y, (x, y) \in [1, 2] \times [1, 2].
$$
\section{绘图结果}
使用P1有限元求解上述问题，得到结果绘制如下：
\begin{figure}[ht]
\centering
\includegraphics[scale=0.5]{./draw/solution.png}
\caption{P1有限元计算结果}
\label{fig:label}
\end{figure}

此外，显然$u(x, y) = \sin  x \sin y$是方程的解析解，绘制如下：
\begin{figure}[ht]
\centering
\includegraphics[scale=0.5]{./draw/trueSolution.png}
\caption{问题的精确解析解}
\label{fig:label}
\end{figure}

\clearpage
\section{问题的误差}
对不同的网格宽度，得到相应的L2误差，如下：

\begin{table}[!htp]
\center
\begin{tabular}{|c|c|c|}
	\hline
	h(网格宽度) & x(单元数量) & L2误差\\
	\hline
	0.25 & 44 & 0.00671632 \\ 
	\hline
	0.2 & 64 & 0.00459713 \\
	\hline
	0.1 & 186 & 0.00169804 \\
	\hline
	0.08 & 394 & 0.000764285 \\
	\hline
	0.07 & 526 & 0.0005646 \\
	\hline
	0.05 & 930 & 0.000304804 \\
	\hline
\end{tabular}
\end{table}

绘制$log(\epsilon)$ - $log(h)$图如下，一次拟合后系数为1.93，大致为2，故误差阶为$O(h^2)$。
\begin{figure}[ht]
\centering
\includegraphics[scale=0.6]{./draw/error.png}
\caption{拟合图}
\label{fig:label}
\end{figure}
\end{document}